\chapter{Dynamic models}

\section{Equation of Motion}

Deriving the \emph{equations of motion} consist in establishing the first order ordinary differential equation or \emph{state space representation} governing the motion of our vehicle, considered as a solid. For aerospace vehicles, classic Newtonian mechanic is generally used and a complete reference on the topic can be found in \cite{tenenbaum_2005}.
The first choice to be made is the determination of state variables. The minimal dimension of the state vector for representing a solid is 12: 3 for the position, 3 for the velocity, 3 for the orientation and 3 for the rotational velocity.


\subsection{Coordinate Frames}
In order to derive the equations of motions, a number of reference coordinate frames must be defined:
\begin{enumerate}
  \item Inertial Frame

  \item Earth Centered Earth Fixed Frame

  \item Body Frame

  \item Navigation Frame (North East Down or local level frame)

  \item Non Singular Navigation Frame (Local level, wander azimuth frame) 

  \item Aerodynamic Frame

  \item Propulsion Frame

\end{enumerate}


